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I'm a beginner. When I searched for results in probabilistic number theory most of the results were asymptotic in nature. Are there any results like with probability 1-$\epsilon$ (w.h.p) some property $P(\epsilon)$ hold? Any references would be also helpful.

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Chebyshev's bias says that there are slightly more non-Pythagorean primes than Pythagorean primes (although the limiting frequency is the same).

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